In many cases, partner reading meant pairing up the children in the classroom. If there was an "odd man out", that lucky child who felt alone discovered he got to read to me. (Sometimes the children would "fight" over who'd be the "leftovers", as they got to calling that one remaining student.)
But every so often a few older kids would show up at my door from one of the upper grade levels and ask if there was anything they could help me with. The upper grades were often rewarded in that manner. Students who'd done their work, or did exceptionally well with something, were allowed to go to other classrooms to offer their assistance. I would take those opportunities to let the kids take turns reading from their readers to those students. The upper grades loved it because it made them feel like teachers.
And once or twice a year I'd arrange for a "read-athon", where an entire upper grade class would come in and pair up to read. The students had a blast doing this, especially if they had an older sibling in that class.